Skip to main content
SpringerLink
Log in

Cubic Bridgeless Graphs and Braces

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

There are many long-standing open problems on cubic bridgeless graphs, for instance, Jaeger’s directed cycle double cover conjecture. On the other hand, many structural properties of braces have been recently discovered. In this work, we bijectively map the cubic bridgeless graphs to braces which we call the hexagon graphs, and explore the structure of hexagon graphs. We show that hexagon graphs are braces that can be generated from the ladder on 8 vertices using two types of McCuaig’s augmentations. In addition, we present a reformulation of Jaeger’s directed cycle double cover conjecture in the class of hexagon graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Jaeger, F.: A survey of the cycle double cover conjecture. In: Alspach, B.R., Godsil, C.D. (eds.) Annals of Discrete Mathematics 27 Cycles in Graphs, volume 115 of North-Holland Mathematics Studies, pp. 1 – 12. North-Holland (1985)

  2. Jiménez, A., Loebl, M.: Directed cycle double covers: fork graphs. arXiv:1310.5539 (2013)

  3. Jiménez, A., Loebl, M.: Directed cycle double covers and cut-obstacles. arXiv:1405.6929 (2014)

  4. Kenyon, R.: The laplacian and dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lovász, L.: Matching structure and the matching lattice. J. Comb. Theory Ser. B 43(2), 187–222 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lovász, L., Plummer, M.D.: Matching Theory. Akadémiai Kiadó, Budapest. Also published as Vol. 121 of the North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam (1986)

  7. McCuaig, W.: Brace generation. J. Graph Theory 38(3), 124–169 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mercat, C.: Discrete riemann surfaces and the ising model. Commun. Math. Phys. 218(1), 177–216 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (MD), London (2001)

    Google Scholar 

  10. Zhang, C.-Q.: Integer Flows and Cycle Covers of Graphs. Marcel Dekker Inc, New York (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CIMFAV, Facultad de Ingeniería, Universidad de Valparaíso, Valparaiso, Chile

    Andrea Jiménez

  2. Institute of Discrete Mathematics, Graz University of Technology, Graz, Austria

    Mihyun Kang

  3. Department of Applied Mathematics, Charles University, Prague, Czech Republic

    Martin Loebl

Authors
  1. Andrea Jiménez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mihyun Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Loebl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrea Jiménez.

Additional information

Andrea Jiménez partially supported by CONICYT: FONDECYT/POSTDOCTORADO 3150673, Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003, Chile, FAPESP (Proc. 2013/03447-6) and CNPq (Proc. 456792/2014-7), Brazil. Mihyun Kang partially supported by the German Research Foundation KA 2748/2-1 and KA 2748/3-1). Martin Loebl partially supported by the Czech Science Foundation under the Contract Number P202-13-21988S.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiménez, A., Kang, M. & Loebl, M. Cubic Bridgeless Graphs and Braces. Graphs and Combinatorics 32, 2473–2495 (2016). https://doi.org/10.1007/s00373-016-1722-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-016-1722-y

Keywords

Search

Navigation